The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory

Abstract

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar N = 4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at N 3 LLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.

Bootstrapping an NMHV amplitude through three loops

“…The same basic assumption for the (parity-even) remainder function R (L) 6 [26,28] results in a consistent solution through four loops [28,29].…”

“…Some additional evidence for factorization beyond NLL was provided in ref. [29], where the fourloop remainder function was computed and found to be consistent with the proposed MRK limit through at least next-to-next-to-leading-logarithmic (NNLL) accuracy.…”

“…In this paper we will follow an alternative approach, the hexagon function bootstrap [26][27][28][29][30]. The philosophy of this program is to bypass integrands altogether and focus on infrared-finite quantities from the very beginning.…”

“…Further perspectives on multi-Regge factorization have been provided by Caron-Huot [55]. The factorization limit has a logarithmic ordering, which allows for the efficient recycling of lower-loop information to higher loops [28,29,56,57]. The recycling is aided by the recognition [56] that in the six-point case the functions relevant for the multi-Regge limit are single-valued harmonic polylogarithms (SVHPLs) [58].…”

“…The number of iterated integrations defines the weight of the hexagon function, which should be 2L for the L-loop contributions to R 6 , V andṼ . Given the hexagon-function ansatz, the near-collinear limit, multi-Regge behavior, and a few other physical constraints uniquely determine the full six-point remainder function at both three [28] and four loops [29]. The uniquenes of the solution, despite the existence of around 6000 unknown parameters in the inital four-loop ansatz, is a testament to the power of the boundary data.…”